Twisted Gan–Gross–Prasad conjecture for biquadratic unitary groups

Let $F$ be a non-Archimedean local field of characteristic $0$, let $E\neq K$ be distinct quadratic extensions of $F$, and put $L=E\otimes_F K$. Let $V$ be an $n$-dimensional skew-Hermitian space over $E$, let $V_K=V\otimes_F K$, and let $\mu$ be a conjugate-symplectic character of $E^\times$. Write

$$m_V(\pi,\mu)=\dim\operatorname{Hom}_{\operatorname{U}(V)}(\pi,\omega_{V,\mu})$$

for $\pi\in\operatorname{Irr}(\operatorname{U}(V_K))$. Fix a trace-zero element $\delta\in E^\times$, set $\psi_{E,\delta}=\psi_F(\operatorname{Tr}_{E/F}(\delta\cdot))$, and write $\operatorname{As}^+_{L/E}$ and $\operatorname{Ind}_L^E$ for the indicated Asai and induction operations. Twisted Gan–Gross–Prasad conjecture. (1) For every $\pi\in\operatorname{Irr}(\operatorname{U}(V_K))$, $m_V(\pi,\mu)\leq 1$. (2) If $M$ is a generic $L$-parameter of $\operatorname{U}(V_K)$ with associated $L$-packet $\Pi_M$, then

$$\sum_V\sum_{\pi\in\Pi_M}m_V(\pi,\mu)=1,$$

where the first sum runs over the two $n$-dimensional skew-Hermitian spaces over $E$. (3) The unique $V_0$ contributing nontrivially is characterized by

$$\epsilon(V_0)=\epsilon\left(\frac12,\operatorname{As}^+_{L/E}(M)\otimes\mu^{-1},\psi_{E,\delta}\right)\cdot\omega_{K/F}(\delta^2)^{n(n-1)/2}.$$

(4) The unique $\pi\in\Pi_M$ contributing nontrivially corresponds, under the LLC with respect to the Whittaker datum associated to $\psi_K$, to the character $\eta$ of $A_M=\prod_{i\in I}\mathbb{Z}/2\mathbb{Z}\cdot a_i$ satisfying

$$\eta(a_i)=\epsilon\left(\frac12,\operatorname{Ind}_L^E\left({}^{\tau}M_i\otimes(M/M_i)\right)\cdot\mu^{-1},\psi_{E,\delta}\right)$$

for each irreducible constituent $M_i$ corresponding to $a_i\in A_M$, equivalently by the second epsilon-factor expression in the source. This conjecture predicts multiplicity one and an explicit description of the unique nonzero multiplicity in each generic packet, including both the relevant Hermitian space and packet member. Its status is not established by the supplied material.